## Polynomial functions of modular lattices

• A polynomial function $$f : L \to L$$ of a lattice $$\mathcal{L}$$ = $$(L; \land, \lor)$$ is generated by the identity function id $$id(x)=x$$ and the constant functions $$c_a (x) = a$$ (for every $$x \in L$$), $$a \in L$$ by applying the operations $$\land, \lor$$ finitely often. Every polynomial function in one or also in several variables is a monotone function of $$\mathcal{L}$$. If every monotone function of $$\mathcal{L}$$is a polynomial function then $$\mathcal{L}$$ is called orderpolynomially complete. In this paper we give a new characterization of finite order-polynomially lattices. We consider doubly irreducible monotone functions and point out their relation to tolerances, especially to central relations. We introduce chain-compatible lattices and show that they have a non-trivial congruence if they contain a finite interval and an infinite chain. The consequences are two new results. A modular lattice $$\mathcal{L}$$ with a finite interval is order-polynomially complete if and only if $$\mathcal{L}$$ is finite projective geometry. If $$\mathcal{L}$$ is simple modular lattice of infinite length then every nontrivial interval is of infinite length and has the same cardinality as any other nontrivial interval of $$\mathcal{L}$$. In the last sections we show the descriptive power of polynomial functions of lattices and present several applications in geometry.